All teachers of trigonometry start with right-angled triangles, and they draw a little square where the 90º angle is. Then they put θ at the other angle, which is the base angle. Then they ask, “Which is the hypotenuse?” A chime of young voices comes back with, “The longest side!” and occasionally, “The side opposite the 90º.”
I know this, because this is what I did for the first five years I taught math.
The title of this article comes from the large number of times that I have heard this question from students and the many times that I have smiled in response. In my mind, trigonometry was beautifully patterned. I did not ever ask the question, “How is it possible that there is always a right angle?” Neither did my students. They were delighted that there was a pattern and that they could extrapolate these patterns to “fake” an answer from it. (Sometimes, they even coloured the sides of the triangle different colours to make it look pretty!)
Then, I travelled through Italy, and saw the Leaning Tower of Pisa and thought, “Aha! I can be adventurous with right-angled triangles now!” and brought it into my classroom. I took a printout of the picture below and handed it to every student and we constructed right-angled triangles around it.
Today, this is not a “rocket-science type lesson plan”, but here is what happened on that day in class. Some started at the right end of the tower and drew a line parallel to the edge of the picture. That ensured a right angle, within reasonable doubt. Some started at the left end of the tower and did not quite see where the line would land if it was perpendicular to the ground.
I heard a couple of them saying, in answer to this possible inaccuracy about the perpendicular,
“Could we run parallel to the wall on the building on the left? That should make it perpendicular, right?”
And someone else said,
“How do you know it is also not leaning? It looks leaning.”
And someone came back with,
“Have you ever heard of the Leaning Big Building of Pisa?” (A little laugh runs around the class.)
“That is the Pisa cathedral,” said the class pedant.
And with each of these comments, I learned a little more about how students learned. And suddenly, I realized that I was a PHYSICS teacher! And we were in the Physics Lab, so I dived for twine and weight bobs, tied some string to each bob and handed them out.
“Does this help to find the perpendicular?”
The author is a teacher who has been in classrooms for over 44 years. She has taught physics and maths in Africa and in India. She can be reached at firstname.lastname@example.org.